| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s + 2·7-s + 9-s + 2·10-s + 11-s + 2·12-s + 4·13-s + 4·14-s + 15-s − 4·16-s + 2·18-s + 19-s + 2·20-s + 2·21-s + 2·22-s + 25-s + 8·26-s + 27-s + 4·28-s − 6·29-s + 2·30-s − 31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 0.755·7-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.577·12-s + 1.10·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.471·18-s + 0.229·19-s + 0.447·20-s + 0.436·21-s + 0.426·22-s + 1/5·25-s + 1.56·26-s + 0.192·27-s + 0.755·28-s − 1.11·29-s + 0.365·30-s − 0.179·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.262025038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.262025038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67700482452114272380308241126, −7.03935937645878056111598934602, −6.17421183843885786351175551715, −5.66828947809040770620230410840, −5.03002634564646730881373580216, −4.05825217259357481442229196233, −3.84304349233068137856219869890, −2.80156791050095622358228961211, −2.11155820434509750665901480807, −1.13466107471059784919116195384,
1.13466107471059784919116195384, 2.11155820434509750665901480807, 2.80156791050095622358228961211, 3.84304349233068137856219869890, 4.05825217259357481442229196233, 5.03002634564646730881373580216, 5.66828947809040770620230410840, 6.17421183843885786351175551715, 7.03935937645878056111598934602, 7.67700482452114272380308241126
